Every infinite-dimensional Hilbert space is real-analytically isomorphic with its unit sphere (Q1908119)

Let \({\mathfrak E}= ({\mathfrak E},|\cdot|)\) be a separable Banach space, \(\omega\) a real analytic (resp., \({\mathfrak C}^p\), \(p=1,2,\dots,\infty\)) norm on \({\mathfrak E}\), and \({\mathfrak E}_0=\ker x^*\) for a \(x^*\neq 0\) in \({\mathfrak E}^*\). The author proves that \({\mathfrak S}_\omega=\{x\in {\mathfrak E}\mid\omega(x)=1\}\) and \({\mathfrak E}_0\) are real-analytically (resp., \({\mathfrak C}^p\)) isomorphic. His result derives the following Theorem: Let \(\mathfrak H\) be an infinite-dimensional Hilbert space, and \(\mathfrak S\) the unit sphere in \(\mathfrak H\). There exists a real-analytic isomorphism \(f\) from \(\mathfrak S\) onto \(\mathfrak H\). His Theorem provides an affirmative answer to the problem by \textit{C. Bessaga} [Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 14, 27-31 (1966; Zbl 0151.17703)]. Next let \(K\) be a compact set in \(\mathfrak S\). As a final Remark, he obtains a \({\mathfrak C}^p\) isomorphism \(f:{\mathfrak E}\backslash K\to{\mathfrak E}\) in the infinite-dimensional separable Banach space \(\mathfrak E\) with \({\mathfrak C}^p\) norm.